Aircraft: B737
Phase of flight: Initial Approach
Anomaly: ATC Issue All Types; Aircraft Equipment Problem Less Severe; Deviation - Track / Heading All Types; Deviation / Discrepancy - Procedural Clearance

We did not believe we were in a deviation. In fact we stated that to the controller when he asked the reason for the 'deviation.' He never disagreed. I am surprised to be filling out this ASAP. Handing off to Denver Approach the controller seemed short with aircraft and a bit testy. A couple of times he was not pleased with aircraft reading back instructions; including us. When he told us to expect the RNAV to RWY 35R the chart had two options for tolerance. A 0.3 and a 0.1. We selected the 0.3 tolerance as we had no ANP predictions (clear day). Also; we are not authorized anything lower than a 0.11. When he cleared for the approach; my new First Officer was a bit unsure about whether we were to turn at MSIMS. I told her that was correct; but I would get clarification from the controller. When I asked; again the controller was short and a bit snooty. Offered no clarification. Only said again that we were cleared for the approach. Just prior to MSIMS our autopilot disconnected. First Officer did a great job executing the turn by hand. I noted; and told her that she is doing fine as the aircraft triangle symbol was still touching the course line; indicating to me that we were within our 0.3 tolerance. About 1/3 of the way through the turn the controller announced we had deviated and to fly heading 180. So we stopped our right turn to final and started a left turn to 180. Within a couple of seconds; he changed his instructions to say fly present heading. So we stopped turn. We were then vectored for the visual. During the vectors; the controller asked the reason for the deviation. I told him that I didn't think we had deviated outside parameters. With no instructions to call anybody after landing; I presumed that he had reflected and perhaps realized that he was maybe spring-loaded to look for problems after not clearing up our earlier question regarding the approach. In my view; he did not realize we were in the turn for final when he announced we were in deviation. I think that he thought we had flown past MSIMS without turning. After he saw we were already in the turn; he amended his instructions to fly present heading. Could he have also been holding us to the 0.1 tolerance rather than the 0.3? He just seemed like he had something else going on that was making him stressed. Perhaps the controller was attempting to hold us to a 0.1 tolerance rather than 0.3. From my perspective; from what I could tell; we were within limits for the approach as the triangle symbol on the NAV display was touching the course. Is it possible the controller was unable to see that we had commenced the turn to final in real time? Again; I had the sense that he felt like we were not turning. If we were in fact in violation; the overall reason is because the autopilot failed when it was time for the right turn to final and we did not react quickly enough. Controllers should try not to be confrontational. Or short. And if pilots ask for clarification; give them some. There are a lot of new people at major airlines nowadays. Aside from that; if there actually was a deviation. It is because the autopilot failed at the worst possible moment. The only real remedy to that would be to require crews to follow through on the controls with the autopilot at an earlier time than is currently required. B737 Captain reported a possible track deviation occurred on arrival into DEN; citing communication difficulties with ATC as contributing.